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The Multiple Permutation Problem and Some Conjectures

In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences' terms, our problem can be really difficult, which calls for novel approaches to work it out for any given pair $(m,n)$. Complete or partial solutions for $m=2, 3$ with smaller $n$'s are listed. Moreover, we have proved the necessary condition for $p(m,n) \neq 0$ and suggested an elegant asymptotic formula for $p(2,n)$. In addition, several challenging conjectures are provided, together with concise comments.